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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.sf_implementation"></a><a class="link" href="sf_implementation.html" title="Additional Implementation Notes">Additional Implementation
    Notes</a>
</h2></div></div></div>
<p>
      The majority of the implementation notes are included with the documentation
      of each function or distribution. The notes here are of a more general nature,
      and reflect more the general implementation philosophy used.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h0"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.implementation_philosophy"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.implementation_philosophy">Implementation
      philosophy</a>
    </h5>
<p>
      "First be right, then be fast."
    </p>
<p>
      There will always be potential compromises to be made between speed and accuracy.
      It may be possible to find faster methods, particularly for certain limited
      ranges of arguments, but for most applications of math functions and distributions,
      we judge that speed is rarely as important as accuracy.
    </p>
<p>
      So our priority is accuracy.
    </p>
<p>
      To permit evaluation of accuracy of the special functions, production of extremely
      accurate tables of test values has received considerable effort.
    </p>
<p>
      (It also required much CPU effort - there was some danger of molten plastic
      dripping from the bottom of JM's laptop, so instead, PAB's Dual-core desktop
      was kept 50% busy for <span class="bold"><strong>days</strong></span> calculating some
      tables of test values!)
    </p>
<p>
      For a specific RealType, say <code class="computeroutput"><span class="keyword">float</span></code>
      or <code class="computeroutput"><span class="keyword">double</span></code>, it may be possible
      to find approximations for some functions that are simpler and thus faster,
      but less accurate (perhaps because there are no refining iterations, for example,
      when calculating inverse functions).
    </p>
<p>
      If these prove accurate enough to be "fit for his purpose", then
      a user may substitute his custom specialization.
    </p>
<p>
      For example, there are approximations dating back from times when computation
      was a <span class="bold"><strong>lot</strong></span> more expensive:
    </p>
<p>
      H Goldberg and H Levine, Approximate formulas for percentage points and normalisation
      of t and chi squared, Ann. Math. Stat., 17(4), 216 - 225 (Dec 1946).
    </p>
<p>
      A H Carter, Approximations to percentage points of the z-distribution, Biometrika
      34(2), 352 - 358 (Dec 1947).
    </p>
<p>
      These could still provide sufficient accuracy for some speed-critical applications.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h1"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.accuracy_and_representation_of_t"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.accuracy_and_representation_of_t">Accuracy
      and Representation of Test Values</a>
    </h5>
<p>
      In order to be accurate enough for as many as possible real types, constant
      values are given to 50 decimal digits if available (though many sources proved
      only accurate near to 64-bit double precision). Values are specified as long
      double types by appending L, unless they are exactly representable, for example
      integers, or binary fractions like 0.125. This avoids the risk of loss of accuracy
      converting from double, the default type. Values are used after <code class="computeroutput"><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="number">1.2345L</span><span class="special">)</span></code> to provide
      the appropriate RealType for spot tests.
    </p>
<p>
      Functions that return constants values, like kurtosis for example, are written
      as
    </p>
<p>
      <code class="computeroutput"><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(-</span><span class="number">3</span><span class="special">)</span> <span class="special">/</span>
      <span class="number">5</span><span class="special">;</span></code>
    </p>
<p>
      to provide the most accurate value that the compiler can compute for the real
      type. (The denominator is an integer and so will be promoted exactly).
    </p>
<p>
      So tests for one third, <span class="bold"><strong>not</strong></span> exactly representable
      with radix two floating-point, (should) use, for example:
    </p>
<p>
      <code class="computeroutput"><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span>
      <span class="number">3</span><span class="special">;</span></code>
    </p>
<p>
      If a function is very sensitive to changes in input, specifying an inexact
      value as input (such as 0.1) can throw the result off by a noticeable amount:
      0.1f is "wrong" by ~1e-7 for example (because 0.1 has no exact binary
      representation). That is why exact binary values - halves, quarters, and eighths
      etc - are used in test code along with the occasional fraction <code class="computeroutput"><span class="identifier">a</span><span class="special">/</span><span class="identifier">b</span></code>
      with <code class="computeroutput"><span class="identifier">b</span></code> a power of two (in order
      to ensure that the result is an exactly representable binary value).
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h2"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.tolerance_of_tests"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.tolerance_of_tests">Tolerance
      of Tests</a>
    </h5>
<p>
      The tolerances need to be set to the maximum of:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Some epsilon value.
        </li>
<li class="listitem">
          The accuracy of the data (often only near 64-bit double).
        </li>
</ul></div>
<p>
      Otherwise when long double has more digits than the test data, then no amount
      of tweaking an epsilon based tolerance will work.
    </p>
<p>
      A common problem is when tolerances that are suitable for implementations like
      Microsoft VS.NET where double and long double are the same size: tests fail
      on other systems where long double is more accurate than double. Check first
      that the suffix L is present, and then that the tolerance is big enough.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h3"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.handling_unsuitable_arguments"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_unsuitable_arguments">Handling
      Unsuitable Arguments</a>
    </h5>
<p>
      In <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1665.pdf" target="_top">Errors
      in Mathematical Special Functions</a>, J. Marraffino &amp; M. Paterno it
      is proposed that signalling a domain error is mandatory when the argument would
      give an mathematically undefined result.
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
          Guideline 1
        </li></ul></div>
<div class="blockquote"><blockquote class="blockquote"><p>
        A mathematical function is said to be defined at a point a = (a1, a2, . .
        .) if the limits as x = (x1, x2, . . .) 'approaches a from all directions
        agree'. The defined value may be any number, or +infinity, or -infinity.
      </p></blockquote></div>
<p>
      Put crudely, if the function goes to + infinity and then emerges 'round-the-back'
      with - infinity, it is NOT defined.
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        The library function which approximates a mathematical function shall signal
        a domain error whenever evaluated with argument values for which the mathematical
        function is undefined.
      </p></blockquote></div>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
          Guideline 2
        </li></ul></div>
<div class="blockquote"><blockquote class="blockquote"><p>
        The library function which approximates a mathematical function shall signal
        a domain error whenever evaluated with argument values for which the mathematical
        function obtains a non-real value.
      </p></blockquote></div>
<p>
      This implementation is believed to follow these proposals and to assist compatibility
      with <span class="emphasis"><em>ISO/IEC 9899:1999 Programming languages - C</em></span> and with
      the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Draft
      Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph
      5</a>. <a class="link" href="error_handling.html" title="Error Handling">See also domain_error</a>.
    </p>
<p>
      See <a class="link" href="pol_ref.html" title="Policy Reference">policy reference</a> for details
      of the error handling policies that should allow a user to comply with any
      of these recommendations, as well as other behaviour.
    </p>
<p>
      See <a class="link" href="error_handling.html" title="Error Handling">error handling</a> for a
      detailed explanation of the mechanism, and <a class="link" href="stat_tut/weg/error_eg.html" title="Error Handling Example">error_handling
      example</a> and <a href="../../../example/error_handling_example.cpp" target="_top">error_handling_example.cpp</a>
    </p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
        If you enable throw but do NOT have try &amp; catch block, then the program
        will terminate with an uncaught exception and probably abort. Therefore to
        get the benefit of helpful error messages, enabling <span class="bold"><strong>all</strong></span>
        exceptions <span class="bold"><strong>and</strong></span> using try&amp;catch is recommended
        for all applications. However, for simplicity, this is not done for most
        examples.
      </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.sf_implementation.h4"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.handling_of_functions_that_are_n"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_of_functions_that_are_n">Handling
      of Functions that are Not Mathematically defined</a>
    </h5>
<p>
      Functions that are not mathematically defined, like the Cauchy mean, fail to
      compile by default. A <a class="link" href="pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">policy</a>
      allows control of this.
    </p>
<p>
      If the policy is to permit undefined functions, then calling them throws a
      domain error, by default. But the error policy can be set to not throw, and
      to return NaN instead. For example,
    </p>
<p>
      <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span>
      <span class="identifier">ignore_error</span></code>
    </p>
<p>
      appears before the first Boost include, then if the un-implemented function
      is called, mean(cauchy&lt;&gt;()) will return std::numeric_limits&lt;T&gt;::quiet_NaN().
    </p>
<div class="warning"><table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
<th align="left">Warning</th>
</tr>
<tr><td align="left" valign="top"><p>
        If <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">has_quiet_NaN</span></code> is false (for example, if
        T is a User-defined type without NaN support), then an exception will always
        be thrown when a domain error occurs. Catching exceptions is therefore strongly
        recommended.
      </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.sf_implementation.h5"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.median_of_distributions"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.median_of_distributions">Median of
      distributions</a>
    </h5>
<p>
      There are many distributions for which we have been unable to find an analytic
      formula, and this has deterred us from implementing <a href="http://en.wikipedia.org/wiki/Median" target="_top">median
      functions</a>, the mid-point in a list of values.
    </p>
<p>
      However a useful numerical approximation for distribution <code class="computeroutput"><span class="identifier">dist</span></code>
      is available as usual as an accessor non-member function median using <code class="computeroutput"><span class="identifier">median</span><span class="special">(</span><span class="identifier">dist</span><span class="special">)</span></code>, that may be evaluated (in the absence of
      an analytic formula) by calling
    </p>
<p>
      <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span></code> (this is the <span class="emphasis"><em>mathematical</em></span>
      definition of course).
    </p>
<p>
      <a href="http://www.amstat.org/publications/jse/v13n2/vonhippel.html" target="_top">Mean,
      Median, and Skew, Paul T von Hippel</a>
    </p>
<p>
      <a href="http://documents.wolfram.co.jp/teachersedition/MathematicaBook/24.5.html" target="_top">Descriptive
      Statistics,</a>
    </p>
<p>
      <a href="http://documents.wolfram.co.jp/v5/Add-onsLinks/StandardPackages/Statistics/DescriptiveStatistics.html" target="_top">and
      </a>
    </p>
<p>
      <a href="http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/NumericalOperationsOnData/3.8.1.html" target="_top">Mathematica
      Basic Statistics.</a> give more detail, in particular for discrete distributions.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h6"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.handling_of_floating_point_infin"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_of_floating_point_infin">Handling
      of Floating-Point Infinity</a>
    </h5>
<p>
      Some functions and distributions are well defined with + or - infinity as argument(s),
      but after some experiments with handling infinite arguments as special cases,
      we concluded that it was generally more useful to forbid this, and instead
      to return the result of <a class="link" href="error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
    </p>
<p>
      Handling infinity as special cases is additionally complicated because, unlike
      built-in types on most - but not all - platforms, not all User-Defined Types
      are specialized to provide <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">infinity</span><span class="special">()</span></code> and would return zero rather than any representation
      of infinity.
    </p>
<p>
      The rationale is that non-finiteness may happen because of error or overflow
      in the users code, and it will be more helpful for this to be diagnosed promptly
      rather than just continuing. The code also became much more complicated, more
      error-prone, much more work to test, and much less readable.
    </p>
<p>
      However in a few cases, for example normal, where we felt it obvious, we have
      permitted argument(s) to be infinity, provided infinity is implemented for
      the <code class="computeroutput"><span class="identifier">RealType</span></code> on that implementation,
      and it is supported and tested by the distribution.
    </p>
<p>
      The range for these distributions is set to infinity if supported by the platform,
      (by testing <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">has_infinity</span></code>) else the maximum value provided
      for the <code class="computeroutput"><span class="identifier">RealType</span></code> by Boost.Math.
    </p>
<p>
      Testing for has_infinity is obviously important for arbitrary precision types
      where infinity makes much less sense than for IEEE754 floating-point.
    </p>
<p>
      So far we have not set <code class="computeroutput"><span class="identifier">support</span><span class="special">()</span></code> function (only range) on the grounds that
      the PDF is uninteresting/zero for infinities.
    </p>
<p>
      Users who require special handling of infinity (or other specific value) can,
      of course, always intercept this before calling a distribution or function
      and return their own choice of value, or other behavior. This will often be
      simpler than trying to handle the aftermath of the error policy.
    </p>
<p>
      Overflow, underflow, denorm can be handled using <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error
      handling policies</a>.
    </p>
<p>
      We have also tried to catch boundary cases where the mathematical specification
      would result in divide by zero or overflow and signalling these similarly.
      What happens at (and near), poles can be controlled through <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error
      handling policies</a>.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h7"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.scale_shape_and_location"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.scale_shape_and_location">Scale, Shape
      and Location</a>
    </h5>
<p>
      We considered adding location and scale to the list of functions, for example:
    </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
<span class="keyword">inline</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">triangular_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">)</span>
<span class="special">{</span>
  <span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">lower</span><span class="special">();</span>
  <span class="identifier">RealType</span> <span class="identifier">mode</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">mode</span><span class="special">();</span>
  <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">upper</span><span class="special">();</span>
  <span class="identifier">RealType</span> <span class="identifier">result</span><span class="special">;</span>  <span class="comment">// of checks.</span>
  <span class="keyword">if</span><span class="special">(</span><span class="keyword">false</span> <span class="special">==</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">check_triangular</span><span class="special">(</span><span class="identifier">BOOST_CURRENT_FUNCTION</span><span class="special">,</span> <span class="identifier">lower</span><span class="special">,</span> <span class="identifier">mode</span><span class="special">,</span> <span class="identifier">upper</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">result</span><span class="special">))</span>
  <span class="special">{</span>
    <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
  <span class="special">}</span>
  <span class="keyword">return</span> <span class="special">(</span><span class="identifier">upper</span> <span class="special">-</span> <span class="identifier">lower</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<p>
      but found that these concepts are not defined (or their definition too contentious)
      for too many distributions to be generally applicable. Because they are non-member
      functions, they can be added if required.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h8"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.notes_on_implementation_of_speci"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.notes_on_implementation_of_speci">Notes
      on Implementation of Specific Functions &amp; Distributions</a>
    </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
          Default parameters for the Triangular Distribution. We are uncertain about
          the best default parameters. Some sources suggest that the Standard Triangular
          Distribution has lower = 0, mode = half and upper = 1. However as a approximation
          for the normal distribution, the most common usage, lower = -1, mode =
          0 and upper = 1 would be more suitable.
        </li></ul></div>
<h5>
<a name="math_toolkit.sf_implementation.h9"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.rational_approximations_used"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">Rational
      Approximations Used</a>
    </h5>
<p>
      Some of the special functions in this library are implemented via rational
      approximations. These are either taken from the literature, or devised by John
      Maddock using <a class="link" href="internals/minimax.html" title="Minimax Approximations and the Remez Algorithm">our Remez code</a>.
    </p>
<p>
      Rational rather than Polynomial approximations are used to ensure accuracy:
      polynomial approximations are often wonderful up to a certain level of accuracy,
      but then quite often fail to provide much greater accuracy no matter how many
      more terms are added.
    </p>
<p>
      Our own approximations were devised either for added accuracy (to support 128-bit
      long doubles for example), or because literature methods were unavailable or
      under non-BSL compatible license. Our Remez code is known to produce good agreement
      with literature results in fairly simple "toy" cases. All approximations
      were checked for convergence and to ensure that they were not ill-conditioned
      (the coefficients can give a theoretically good solution, but the resulting
      rational function may be un-computable at fixed precision).
    </p>
<p>
      Recomputing using different Remez implementations may well produce differing
      coefficients: the problem is well known to be ill conditioned in general, and
      our Remez implementation often found a broad and ill-defined minima for many
      of these approximations (of course for simple "toy" examples like
      approximating <code class="computeroutput"><span class="identifier">exp</span></code> the minima
      is well defined, and the coefficients should agree no matter whose Remez implementation
      is used). This should not in general effect the validity of the approximations:
      there's good literature supporting the idea that coefficients can be "in
      error" without necessarily adversely effecting the result. Note that "in
      error" has a special meaning in this context, see <a href="http://front.math.ucdavis.edu/0101.5042" target="_top">"Approximate
      construction of rational approximations and the effect of error autocorrection.",
      Grigori Litvinov, eprint arXiv:math/0101042</a>. Therefore the coefficients
      still need to be accurately calculated, even if they can be in error compared
      to the "true" minimax solution.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h10"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.representation_of_mathematical_c"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.representation_of_mathematical_c">Representation
      of Mathematical Constants</a>
    </h5>
<p>
      A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used to provide high
      accuracy constants to mathematical functions and distributions, since it is
      important to provide values uniformly for both built-in float, double and long
      double types, and for User Defined types in <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
      like <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>.
      and others like NTL::quad_float and NTL::RR.
    </p>
<p>
      To permit calculations in this Math ToolKit and its tests, (and elsewhere)
      at about 100 decimal digits with NTL::RR type, it is obviously necessary to
      define constants to this accuracy.
    </p>
<p>
      However, some compilers do not accept decimal digits strings as long as this.
      So the constant is split into two parts, with the 1st containing at least long
      double precision, and the 2nd zero if not needed or known. The 3rd part permits
      an exponent to be provided if necessary (use zero if none) - the other two
      parameters may only contain decimal digits (and sign and decimal point), and
      may NOT include an exponent like 1.234E99 (nor a trailing F or L). The second
      digit string is only used if T is a User-Defined Type, when the constant is
      converted to a long string literal and lexical_casted to type T. (This is necessary
      because you can't use a numeric constant since even a long double might not
      have enough digits).
    </p>
<p>
      For example, pi is defined:
    </p>
<pre class="programlisting"><span class="identifier">BOOST_DEFINE_MATH_CONSTANT</span><span class="special">(</span><span class="identifier">pi</span><span class="special">,</span>
  <span class="number">3.141592653589793238462643383279502884197169399375105820974944</span><span class="special">,</span>
  <span class="number">5923078164062862089986280348253421170679821480865132823066470938446095505</span><span class="special">,</span>
  <span class="number">0</span><span class="special">)</span>
</pre>
<p>
      And used thus:
    </p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">;</span>

<span class="keyword">double</span> <span class="identifier">diameter</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">radius</span> <span class="special">=</span> <span class="identifier">diameter</span> <span class="special">*</span> <span class="identifier">pi</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>

<span class="keyword">or</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">&gt;()</span>
</pre>
<p>
      Note that it is necessary (if inconvenient) to specify the type explicitly.
    </p>
<p>
      So you cannot write
    </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;&gt;();</span>  <span class="comment">// could not deduce template argument for 'T'</span>
</pre>
<p>
      Neither can you write:
    </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span> <span class="comment">// Context does not allow for disambiguation of overloaded function</span>
<span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">();</span> <span class="comment">// Context does not allow for disambiguation of overloaded function</span>
</pre>
<h5>
<a name="math_toolkit.sf_implementation.h11"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.thread_safety"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.thread_safety">Thread
      safety</a>
    </h5>
<p>
      Reporting of error by setting <code class="computeroutput"><span class="identifier">errno</span></code>
      should be thread-safe already (otherwise none of the std lib math functions
      would be thread safe?). If you turn on reporting of errors via exceptions,
      <code class="computeroutput"><span class="identifier">errno</span></code> gets left unused anyway.
    </p>
<p>
      For normal C++ usage, the Boost.Math <code class="computeroutput"><span class="keyword">static</span>
      <span class="keyword">const</span></code> constants are now thread-safe
      so for built-in real-number types: <code class="computeroutput"><span class="keyword">float</span></code>,
      <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span>
      <span class="keyword">double</span></code> are all thread safe.
    </p>
<p>
      For User_defined types, for example, <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>,
      the Boost.Math should also be thread-safe, (thought we are unsure how to rigorously
      prove this).
    </p>
<p>
      (Thread safety has received attention in the C++11 Standard revision, so hopefully
      all compilers will do the right thing here at some point.)
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h12"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.sources_of_test_data"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.sources_of_test_data">Sources
      of Test Data</a>
    </h5>
<p>
      We found a large number of sources of test data. We have assumed that these
      are <span class="emphasis"><em>"known good"</em></span> if they agree with the results
      from our test and only consulted other sources for their <span class="emphasis"><em>'vote'</em></span>
      in the case of serious disagreement. The accuracy, actual and claimed, vary
      very widely. Only <a href="http://functions.wolfram.com/" target="_top">Wolfram Mathematica
      functions</a> provided a higher accuracy than C++ double (64-bit floating-point)
      and was regarded as the most-trusted source by far. The <a href="http://www.r-project.org/" target="_top">The
      R Project for Statistical Computing</a> provided the widest range of distributions,
      but the usual Intel X86 distribution uses 64-but doubles, so our use was limited
      to the 15 to 17 decimal digit accuracy.
    </p>
<p>
      A useful index of sources is: <a href="http://www.sal.hut.fi/Teaching/Resources/ProbStat/table.html" target="_top">Web-oriented
      Teaching Resources in Probability and Statistics</a>
    </p>
<p>
      <a href="http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm" target="_top">Statlet</a>:
      Is a Javascript application that calculates and plots probability distributions,
      and provides the most complete range of distributions:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        Bernoulli, Binomial, discrete uniform, geometric, hypergeometric, negative
        binomial, Poisson, beta, Cauchy-Lorentz, chi-squared, Erlang, exponential,
        extreme value, Fisher, gamma, Laplace, logistic, lognormal, normal, Pareto,
        Student's t, triangular, uniform, and Weibull.
      </p></blockquote></div>
<p>
      It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, &amp; critical
      values for 5 tail values.
    </p>
<p>
      It is also the only independent source found for the Weibull distribution;
      unfortunately it appears to suffer from very poor accuracy in areas where the
      underlying special function is known to be difficult to implement.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h13"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.testing_for_invalid_parameters_t"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.testing_for_invalid_parameters_t">Testing
      for Invalid Parameters to Functions and Constructors</a>
    </h5>
<p>
      After finding that some 'bad' parameters (like NaN) were not throwing a <code class="computeroutput"><span class="identifier">domain_error</span></code> exception as they should, a
      function
    </p>
<p>
      <code class="computeroutput"><span class="identifier">check_out_of_range</span></code> (in <code class="computeroutput"><span class="identifier">test_out_of_range</span><span class="special">.</span><span class="identifier">hpp</span></code>) was devised by JM to check (using Boost.Test's
      BOOST_CHECK_THROW macro) that bad parameters passed to constructors and functions
      throw <code class="computeroutput"><span class="identifier">domain_error</span></code> exceptions.
    </p>
<p>
      Usage is <code class="computeroutput"><span class="identifier">check_out_of_range</span><span class="special">&lt;</span> <span class="identifier">DistributionType</span>
      <span class="special">&gt;(</span><span class="identifier">list</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">params</span><span class="special">);</span></code>
      Where list-of-params is a list of <span class="bold"><strong>valid</strong></span> parameters
      from which the distribution can be constructed - ie the same number of args
      are passed to the function, as are passed to the distribution constructor.
    </p>
<p>
      The values of the parameters are not important, but must be <span class="bold"><strong>valid</strong></span>
      to pass the constructor checks; the default values are suitable, but must be
      explicitly provided, for example:
    </p>
<pre class="programlisting"><span class="identifier">check_out_of_range</span><span class="special">&lt;</span><span class="identifier">extreme_value_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;</span> <span class="special">&gt;(</span><span class="number">1</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span>
</pre>
<p>
      Checks made are:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Infinity or NaN (if available) passed in place of each of the valid params.
        </li>
<li class="listitem">
          Infinity or NaN (if available) as a random variable.
        </li>
<li class="listitem">
          Out-of-range random variable passed to pdf and cdf (ie outside of "range(DistributionType)").
        </li>
<li class="listitem">
          Out-of-range probability passed to quantile function and complement.
        </li>
</ul></div>
<p>
      but does <span class="bold"><strong>not</strong></span> check finite but out-of-range
      parameters to the constructor because these are specific to each distribution,
      for example:
    </p>
<pre class="programlisting"><span class="identifier">BOOST_CHECK_THROW</span><span class="special">(</span><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">pareto_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">),</span> <span class="number">0</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">);</span>
<span class="identifier">BOOST_CHECK_THROW</span><span class="special">(</span><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">pareto_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">),</span> <span class="number">0</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">);</span>
</pre>
<p>
      checks <code class="computeroutput"><span class="identifier">scale</span></code> and <code class="computeroutput"><span class="identifier">shape</span></code> parameters are both &gt; 0 by checking
      that <code class="computeroutput"><span class="identifier">domain_error</span></code> exception
      is thrown if either are == 0.
    </p>
<p>
      (Use of <code class="computeroutput"><span class="identifier">check_out_of_range</span></code>
      function may mean that some previous tests are now redundant).
    </p>
<p>
      It was also noted that if more than one parameter is bad, then only the first
      detected will be reported by the error message.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h14"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.creating_and_managing_the_equati"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.creating_and_managing_the_equati">Creating
      and Managing the Equations</a>
    </h5>
<p>
      Equations that fit on a single line can most easily be produced by inline Quickbook
      code using templates for Unicode Greek and Unicode Math symbols. All Greek
      letter and small set of Math symbols is available at /boost-path/libs/math/doc/sf_and_dist/html4_symbols.qbk
    </p>
<p>
      Where equations need to use more than one line, real Math editors were used.
    </p>
<p>
      The primary source for the equations is now <a href="http://www.w3.org/Math/" target="_top">MathML</a>:
      see the *.mml files in libs/math/doc/sf_and_dist/equations/.
    </p>
<p>
      These are most easily edited by a GUI editor such as <a href="http://mathcast.sourceforge.net/home.html" target="_top">Mathcast</a>,
      please note that the equation editor supplied with Open Office currently mangles
      these files and should not currently be used.
    </p>
<p>
      Conversion to SVG was achieved using <a href="https://sourceforge.net/projects/svgmath/" target="_top">SVGMath</a>
      and a command line such as:
    </p>
<pre class="programlisting">$for file in *.mml; do
&gt;/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \
&gt;&gt;$file &gt; $(basename $file .mml).svg
&gt;done
</pre>
<p>
      See also the section on "Using Python to run Inkscape" and "Using
      inkscape to convert scalable vector SVG files to Portable Network graphic PNG".
    </p>
<p>
      Note that SVGMath requires that the mml files are <span class="bold"><strong>not</strong></span>
      wrapped in an XHTML XML wrapper - this is added by Mathcast by default - one
      workaround is to copy an existing mml file and then edit it with Mathcast:
      the existing format should then be preserved. This is a bug in the XML parser
      used by SVGMath which the author is aware of.
    </p>
<p>
      If necessary the XHTML wrapper can be removed with:
    </p>
<pre class="programlisting">cat filename | tr -d "\r\n" | sed -e 's/.*\(&lt;math[^&gt;]*&gt;.*&lt;/math&gt;\).*/\1/' &gt; newfile</pre>
<p>
      Setting up fonts for SVGMath is currently rather tricky, on a Windows XP system
      JM's font setup is the same as the sample config file provided with SVGMath
      but with:
    </p>
<pre class="programlisting">    &lt;!-- Double-struck --&gt;
    &lt;mathvariant name="double-struck" family="Mathematica7, Lucida Sans Unicode"/&gt;
</pre>
<p>
      changed to:
    </p>
<pre class="programlisting">    &lt;!-- Double-struck --&gt;
    &lt;mathvariant name="double-struck" family="Lucida Sans Unicode"/&gt;
</pre>
<p>
      Note that unlike the sample config file supplied with SVGMath, this does not
      make use of the <a href="http://support.wolfram.com/technotes/fonts/windows/latestfonts.html" target="_top">Mathematica
      7 font</a> as this lacks sufficient Unicode information for it to be used
      with either SVGMath or XEP "as is".
    </p>
<p>
      Also note that the SVG files in the repository are almost certainly Windows-specific
      since they reference various Windows Fonts.
    </p>
<p>
      PNG files can be created from the SVGs using <a href="http://xmlgraphics.apache.org/batik/tools/rasterizer.html" target="_top">Batik</a>
      and a command such as:
    </p>
<pre class="programlisting">java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg</pre>
<p>
      Or using Inkscape (File, Export bitmap, Drawing tab, bitmap size (default size,
      100 dpi), Filename (default). png)
    </p>
<p>
      or Using Cygwin, a command such as:
    </p>
<pre class="programlisting">for file in *.svg; do
  /cygdrive/c/progra~1/Inkscape/inkscape -d 120 -e $(cygpath -a -w $(basename $file .svg).png) $(cygpath -a -w $file);
done</pre>
<p>
      Using BASH
    </p>
<pre class="programlisting"># Convert single SVG to PNG file.
# /c/progra~1/Inkscape/inkscape -d 120 -e a.png a.svg
</pre>
<p>
      or to convert All files in folder SVG to PNG.
    </p>
<pre class="programlisting">for file in *.svg; do
/c/progra~1/Inkscape/inkscape -d 120 -e $(basename $file .svg).png $file
done
</pre>
<p>
      Currently Inkscape seems to generate the better looking PNGs.
    </p>
<p>
      The PDF is generated into \pdf\math.pdf using a command from a shell or command
      window with current directory \math_toolkit\libs\math\doc\sf_and_dist, typically:
    </p>
<pre class="programlisting">bjam -a pdf &gt;math_pdf.log</pre>
<p>
      Note that XEP will have to be configured to <span class="bold"><strong>use and embed</strong></span>
      whatever fonts are used by the SVG equations (almost certainly editing the
      sample xep.xml provided by the XEP installation). If you fail to do this you
      will get XEP warnings in the log file like
    </p>
<pre class="programlisting">[warning]could not find any font family matching "Times New Roman"; replaced by Helvetica</pre>
<p>
      (html is the default so it is generated at libs\math\doc\html\index.html using
      command line &gt;bjam -a &gt; math_toolkit.docs.log).
    </p>
<pre class="programlisting"><span class="special">&lt;!--</span> <span class="identifier">Sample</span> <span class="identifier">configuration</span> <span class="keyword">for</span> <span class="identifier">Windows</span> <span class="identifier">TrueType</span> <span class="identifier">fonts</span><span class="special">.</span>  <span class="special">--&gt;</span>
</pre>
<p>
      is provided in the xep.xml downloaded, but the Windows TrueType fonts are commented
      out.
    </p>
<p>
      JM's XEP config file \xep\xep.xml has the following font configuration section
      added:
    </p>
<pre class="programlisting">    &lt;font-group xml:base="file:/C:/Windows/Fonts/" label="Windows TrueType" embed="true" subset="true"&gt;
      &lt;font-family name="Arial"&gt;
        &lt;font&gt;&lt;font-data ttf="arial.ttf"/&gt;&lt;/font&gt;
        &lt;font style="oblique"&gt;&lt;font-data ttf="ariali.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="arialbd.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold" style="oblique"&gt;&lt;font-data ttf="arialbi.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

      &lt;font-family name="Times New Roman" ligatures="&amp;#xFB01; &amp;#xFB02;"&gt;
        &lt;font&gt;&lt;font-data ttf="times.ttf"/&gt;&lt;/font&gt;
        &lt;font style="italic"&gt;&lt;font-data ttf="timesi.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="timesbd.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold" style="italic"&gt;&lt;font-data ttf="timesbi.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

      &lt;font-family name="Courier New"&gt;
        &lt;font&gt;&lt;font-data ttf="cour.ttf"/&gt;&lt;/font&gt;
        &lt;font style="oblique"&gt;&lt;font-data ttf="couri.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="courbd.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold" style="oblique"&gt;&lt;font-data ttf="courbi.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

      &lt;font-family name="Tahoma" embed="true"&gt;
        &lt;font&gt;&lt;font-data ttf="tahoma.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="tahomabd.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

      &lt;font-family name="Verdana" embed="true"&gt;
        &lt;font&gt;&lt;font-data ttf="verdana.ttf"/&gt;&lt;/font&gt;
        &lt;font style="oblique"&gt;&lt;font-data ttf="verdanai.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="verdanab.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold" style="oblique"&gt;&lt;font-data ttf="verdanaz.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

      &lt;font-family name="Palatino" embed="true" ligatures="&amp;#xFB00; &amp;#xFB01; &amp;#xFB02; &amp;#xFB03; &amp;#xFB04;"&gt;
        &lt;font&gt;&lt;font-data ttf="pala.ttf"/&gt;&lt;/font&gt;
        &lt;font style="italic"&gt;&lt;font-data ttf="palai.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold"&gt;&lt;font-data ttf="palab.ttf"/&gt;&lt;/font&gt;
        &lt;font weight="bold" style="italic"&gt;&lt;font-data ttf="palabi.ttf"/&gt;&lt;/font&gt;
      &lt;/font-family&gt;

    &lt;font-family name="Lucida Sans Unicode"&gt;
         &lt;!-- &lt;font&gt;&lt;font-data ttf="lsansuni.ttf"&gt;&lt;<span class="emphasis"><em>font&gt; --&gt;
         &lt;!-- actually called l_10646.ttf on Windows 2000 and Vista Sp1 --&gt;
         &lt;font&gt;&lt;font-data ttf="l_10646.ttf"</em></span>&gt;&lt;/font&gt;
    &lt;/font-family&gt;
</pre>
<p>
      PAB had to alter his because the Lucida Sans Unicode font had a different name.
      Other changes are very likely to be required if you are not using Windows.
    </p>
<p>
      XZ authored his equations using the venerable Latex, JM converted these to
      MathML using <a href="http://gentoo-wiki.com/HOWTO_Convert_LaTeX_to_HTML_with_MathML" target="_top">mxlatex</a>.
      This process is currently unreliable and required some manual intervention:
      consequently Latex source is not considered a viable route for the automatic
      production of SVG versions of equations.
    </p>
<p>
      Equations are embedded in the quickbook source using the <span class="emphasis"><em>equation</em></span>
      template defined in math.qbk. This outputs Docbook XML that looks like:
    </p>
<pre class="programlisting">&lt;inlinemediaobject&gt;
&lt;imageobject role="html"&gt;
&lt;imagedata fileref="../equations/myfile.png"&gt;&lt;/imagedata&gt;
&lt;/imageobject&gt;
&lt;imageobject role="print"&gt;
&lt;imagedata fileref="../equations/myfile.svg"&gt;&lt;/imagedata&gt;
&lt;/imageobject&gt;
&lt;/inlinemediaobject&gt;
</pre>
<p>
      MathML is not currently present in the Docbook output, or in the generated
      HTML: this needs further investigation.
    </p>
<h5>
<a name="math_toolkit.sf_implementation.h15"></a>
      <span class="phrase"><a name="math_toolkit.sf_implementation.producing_graphs"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.producing_graphs">Producing
      Graphs</a>
    </h5>
<p>
      Graphs were produced in SVG format and then converted to PNG's using the same
      process as the equations.
    </p>
<p>
      The programs <code class="computeroutput"><span class="special">/</span><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">doc</span><span class="special">/</span><span class="identifier">sf_and_dist</span><span class="special">/</span><span class="identifier">graphs</span><span class="special">/</span><span class="identifier">dist_graphs</span><span class="special">.</span><span class="identifier">cpp</span></code> and <code class="computeroutput"><span class="special">/</span><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">doc</span><span class="special">/</span><span class="identifier">sf_and_dist</span><span class="special">/</span><span class="identifier">graphs</span><span class="special">/</span><span class="identifier">sf_graphs</span><span class="special">.</span><span class="identifier">cpp</span></code> generate
      the SVG's directly using the <a href="http://code.google.com/soc/2007/boost/about.html" target="_top">Google
      Summer of Code 2007</a> project of Jacob Voytko (whose work so far, considerably
      enhanced and now reasonably mature and usable, by Paul A. Bristow, is at .\boost-sandbox\SOC\2007\visualization).
    </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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